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Properties

Label 2-975-5.4-c1-0-34
Degree $2$
Conductor $975$
Sign $-0.894 - 0.447i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.21i·2-s − i·3-s + 0.525·4-s − 1.21·6-s − 2.59i·7-s − 3.06i·8-s − 9-s − 6.11·11-s − 0.525i·12-s + i·13-s − 3.14·14-s − 2.67·16-s + 4.37i·17-s + 1.21i·18-s − 4.14·19-s + ⋯
L(s)  = 1  − 0.858i·2-s − 0.577i·3-s + 0.262·4-s − 0.495·6-s − 0.979i·7-s − 1.08i·8-s − 0.333·9-s − 1.84·11-s − 0.151i·12-s + 0.277i·13-s − 0.841·14-s − 0.668·16-s + 1.06i·17-s + 0.286i·18-s − 0.951·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.257130 + 1.08922i\)
\(L(\frac12)\) \(\approx\) \(0.257130 + 1.08922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 - iT \)
good2 \( 1 + 1.21iT - 2T^{2} \)
7 \( 1 + 2.59iT - 7T^{2} \)
11 \( 1 + 6.11T + 11T^{2} \)
17 \( 1 - 4.37iT - 17T^{2} \)
19 \( 1 + 4.14T + 19T^{2} \)
23 \( 1 + 7.95iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 5.36T + 31T^{2} \)
37 \( 1 - 6.90iT - 37T^{2} \)
41 \( 1 + 9.19T + 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 + 1.21iT - 47T^{2} \)
53 \( 1 + 4.95iT - 53T^{2} \)
59 \( 1 + 5.44T + 59T^{2} \)
61 \( 1 - 9.99T + 61T^{2} \)
67 \( 1 + 4.87iT - 67T^{2} \)
71 \( 1 - 1.39T + 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 + 8.04T + 89T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.08309991423177515440148232323, −8.488166314149629556539657188314, −7.976716645383903155291746106577, −6.86302423011086710225517841180, −6.39666038891798690009618866808, −4.97382454918936857364315590046, −3.92245141891678399505714291558, −2.77393800662289670775833253909, −1.93155707323289424610386681485, −0.46588419293395909142308363440, 2.33599594096713462151716379473, 3.02713081808241100620980808319, 4.78267440103099686365087929391, 5.39942255363953537567862071304, 6.03299893498969028725903585812, 7.19037355293932616837919141796, 7.988462986162453339607033704875, 8.583435862691634577139682760458, 9.598174573123218307380655626523, 10.43227411867876251174123221279

Graph of the $Z$-function along the critical line